3 Answers
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Moonlight is not almost as bright as mid-dawn. Moonlight is really fairly low illumination, and human eyes don't detect color well in low illumination.

Moonlight is reflected sunlight. If there's enough of it, it produces color mostly the same way sunlight does. This page, found through a Google search, shows a spectrum of moonlight. All the colors are there. There simply isn't enough moonlight for our eyes to detect color well.

The "dark limit of civil twilight" corresponds to ten times as much luminous power per unit area as a full moon. If you look at the Wikipedia article for what counts as twilight, the pictures there aren't very colorful.

It's also important to note that, contrary to popular belief, sunlight and moonlight are actually the same color. A lot of people think that the sun produces yellow light, which isn't really true. The light from the sun is pure white. You can get a good idea of what color it is by looking at clouds, which are nearly perfect scattering surfaces in the visible spectrum. You can't look at the sun itself, of course. If it's overhead (where atmospheric effects don't change it's color) then it's way too bright to get a good look at.
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Colin KSep 28 '11 at 21:47

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@Colin: It's important to note that your definition of pure white is one of human perception (and that does make perfect sense, especially in the context of the question). That said, if one simply wants to describe the black-body spectrum, it'd be reasonable to say that it's yellow - the peak of the spectrum is there.
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JefromiSep 29 '11 at 0:42

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@Colin: I didn't mean to be mocking - I was careful to say your definition made perfect sense! My point was just to clarify a little bit what you meant, since this is a very popular science kind of question, which many people without much science background might see, and they might benefit from knowing that the sun's spectrum is in some sense yellow (though it's white to our eyes, as you said).
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JefromiSep 29 '11 at 2:12

@Vineet the blackbody radiation does not correspond to yellow. It has all wavelengths, but it has slightly more yellowish content than other visible colours. A white surface scatters all incoming lights. Since the Sun has been the dominant light source on Earth, what we see in white colour is Sun's spectrum. So if you want to design a lamp that looks perfectly white, it must be indistinguishable to your eyes from sunlight.
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timurOct 1 '11 at 4:02

'Color' is just the human minds way of interpreting wavelength. Your eyes have two types of receptors. Without going into too much detail, one (rods) 'sees' intensity and the other (cones) 'sees' intensity and wavelength - that is, color. The rods are more sensitive than the cones. Below certain intensity the cones ability to detect color simply doesn't work. Moonlight falls well above the rods ability to see, but below the cones ability to register wavelength. In other words, you are color blind in low light. But the colors are still there.

Stepping down in luminosity, go from the outdoors in sunlight, move indoors to a room with the light on, then step outside into you front yard with a full moon. In each of these cases, even though eventually the environment seems quite bright, it still takes time for your eyes to adapt to the lower light levels.

I have seen moonlit nights here (Australia) where some colours, mainly in the green areas of the spectrum, are almost clear; the amount of light in the environment however is still much lower than is available at dawn. And of course in low light conditions our eyes loose the ability to distinguish colours as the cells in play (the cones) need a high level of light to work; the cells that work in low light (the rods) only see greys.